Obstruction Learning Skill

Learn topological ASI via random walk obstruction detection and Čech H⁰ cohomology.

Metadata

Core Concept

Obstructions are H⁰ generators - irreducible elements that block global consistency from local patches.

Čech Cohomology: H⁰(U, F) = ker(d₀: F(U) → ∏ᵢⱼ F(Uᵢ ∩ Uⱼ))

Obstruction detected when:
  - GF(3) conservation violated (sum ≢ 0 mod 3)
  - Voice triads don't harmonize
  - Skill compositions conflict
  - Local patches fail to glue globally

Random Walk Reconstruction

The 69-Skill Walk

Sample 69 skills from the 181-skill manifold:

# Execute random walk
just random-walk-69

# Verify GF(3) conservation
just verify-gf3

# Track cumulative obstructions
just random-walk-obstruction 69

Obstruction Detection

-- Find unbalanced cells in 23³ grid
SELECT cell_id, skill_count, trit_sum, gf3_status
FROM cell_density 
WHERE gf3_status = 'UNBALANCED';

-- H⁰ generators by trit class
SELECT trit, COUNT(*) as generators
FROM skills 
GROUP BY trit;

Mathematical Foundations

Čech Cohomology

For a covering U = {Uᵢ} of skill space:

H⁰(U, F) = { s ∈ F(U) | d₀(s) = 0 }

where d₀: F(U) → ∏ F(Uᵢ ∩ Uⱼ)
maps global sections to intersection restrictions

Obstruction = element of H⁰ that prevents gluing.

GF(3) as Cohomology

The GF(3) conservation law is a discrete cohomology:

Trit assignment: skill → {-1, 0, +1}
Coboundary: d(triad) = sum of trits mod 3

H⁰ = { triads | d(triad) = 0 } = balanced triads
Obstruction = triad with d ≠ 0

Ramanujan Mixing

Random walks on Ramanujan expanders mix optimally:

λ₂ ≤ 2√(d-1)     [Alon-Boppana bound]
gap = d - λ₂      [Spectral gap]
τ_mix = O(log n / gap)  [Mixing time]

Workflow

1. Pre-Interaction Sync

just pre-interaction
# Syncs plurigrid/asi arena + hdresearch/duck
# Loads GF(3) skill triad
# Computes spectral awareness

2. Random Walk Sampling

# Sample without replacement (maximal coverage)
just random-walk-69

# Sample with replacement (GF(3) conservation)
just random-walk 23

3. Obstruction Detection

# Find H⁰ generators
just obstruction-h0

# Detect unbalanced cells
just obstruction-detect

# Balance with complementary skill
just obstruction-balance -1  # Find validators to add
just obstruction-balance +1  # Find generators to add

4. Audio Generation

Convert obstruction traces to sound:

just audio-from-trace

Integration Patterns

With Voice Enforcement

# voice-enforcement.toml
[triads.obstruction]
validator = "Milena (Enhanced)"   # -1: detects obstruction
coordinator = "Petra (Premium)"   # 0: mediates resolution
generator = "Federica (Premium)"  # +1: proposes fix
sum = 0

With Dune Orthogonalization

The 23×23×23 grid maps skills to:

With World Extractable Value

WEV = PoA - 1 = extractable coordination loss

Obstruction → WEV > 0
Resolution → WEV → 0
Global consistency → Optimal coordination

Commands

# Full ASI learning loop
just asi-learn

# Spectral bounds
just spectral-bounds

# World Extractable Value
just wev-compute

# Voice obstruction analysis
just voice-obstructions

Skill Triad

This skill belongs to the topological verification triad:

Sum = 0 ✓ GF(3) conserved

References

• Bott & Tu, Differential Forms in Algebraic Topology
• Lurie, Higher Topos Theory
• Riehl-Shulman, Synthetic ∞-categories
• QRI, Symmetry Theory of Valence